Time in model theory
The time order between interpretations of expressions
Given a model, expressions do not receive their interpretations all
at once, but only the ones after the others, because these
interpretations depend on each other, thus must be calculated after
each other. This time order of interpretation between expressions,
follows the hierarchical order from sub-expressions to expressions
Take for example, the formula xy+x=3. In order for it
to make sense, the variables x and y must take a
value first. Then, xy takes a value, obtained by multiplying
the values of x and y. Then, xy+x
takes a value based on the previous ones. Then, the whole formula
(xy+x=3) takes a Boolean value (true or false).
But this value depends on those of the free variables x and
y. Finally, taking for example the ground formula ∀x,
∃y, xy+x=3, its Boolean value (which is false
in the world of real numbers), «is calculated from» those taken by
the previous formula for all possible values of x and y,
and therefore comes after them.
A finite list of formulas in a theory may be interpreted by a single
big formula containing them all. This only requires to successively
integrate (or describe) all individual formulas from the list in the
big one, with no need to represent formulas as objects (values of a
variable). This big formula comes (is interpreted) after them all,
but still belongs to the same theory. But for only one formula to
describe the interpretation of an infinity of formulas (such as all
possible formulas, handled as values of a variable), would require
to switch to the framework of one-model theory.
The metaphor of the usual time
I can speak of «what I told about at that time»: it has a sense if
that past saying had one, as I got that meaning and I remember it.
But mentioning «what I mean», would not itself inform on what it is,
as it might be anything, and becomes absurd in a phrase that
modifies or contradicts this meaning («the opposite of what I'm
saying»). Mentioning «what I will mention tomorrow», even if I knew
what I will say, would not suffice to already provide its meaning
either: in case I will mention «what I told about yesterday» (thus
now) it would make a vicious circle; but even if the form of my
ensured that its meaning will exist tomorrow, this would still not
provide it today. I might try to speculate on it, but the actual
meaning of future statements will only arise once actually expressed
in context. By lack of interest to describe phrases without their
meaning, we should rather restrict our study to past expressions,
while just "living" the present ones and ignoring future ones.
So, my current universe of the past that I can describe today,
includes the one of yesterday, but also my yesterday's comments
about it and their meaning. I can thus describe today things outside
the universe I could describe yesterday. Meanwhile I neither learned
to speak Martian nor acquired a new transcendental intelligence, but
the same language applies to a broader universe with new objects. As
these new objects are of the same kinds as the old ones, my universe
of today may look similar to that of yesterday; but from one
universe to another, the same expressions can take different
Like historians, mathematical theories can only «at every given
time» describe a universe of past mathematical objects, while this
interpretation itself «happens» in a mathematical present outside
As a one-model
theory T' describes a theory T with a model M,
the components (notions and structures)
of the model [T,M] of T', actually fall into 3
Even if describing «the universe of all mathematical objects»
(model of set theory), means describing everything, this
«everything» that is described, is only at any time the current
universe, the one of our past ; our act of interpreting
expressions there, forms our present beyond this past. And then,
describing our previous act of description, means adding to this
previous description (this «everything» described) something else
This last part of [T,M] is a mathematical construction
determined by the combination of both systems T and M
but it is not directly contained in them : it is built after
- The components of T and its developments as a formal
system (abstract types, structure symbols, expressions, axioms,
proofs from axioms), that aim to describe the model but remain
outside it and independent of it.
- The components of M (interpretations of types and
- The interpretation (attribution of values) of all expressions
of T in M, for any values of their free
So, the model [T,M] of T', encompassing the
present theory T with the interpretation of all its formulas
in the present universe M of past objects, is the next universe
of the past, which will come as the infinity of all current
interpretations (in M) of formulas of T will become
Or can it be otherwise ? Would it be possible for a theory T
to express or simulate the notion of its own formulas and
compute their values ?
in 1.7, some theories (such as model theory, and set theory
from which it can be developed) are actually able to describe
themselves: they can describe in each model a system looking like
a copy of the same theory, with a notion of "all its formulas"
(including objects that are copies of its own formulas). However
then, according to the Truth Undefinability
theorem, no single formula (invariant predicate) can ever
give the correct boolean values to all object copies of ground
formulas, in conformity with the values of these formulas in the
The Berry paradox
This famous paradox is the idea of "defining" a natural number n
as "the smallest number not definable in less than twenty words".
This would define in 10 words a number... not definable in less than
20 words. But this does not bring a contradiction in mathematics
because it is not a mathematical definition. By making it more
precise, we can form a simple proof of the truth undefinability
theorem (but not a fully rigorous one):
Let us assume a fixed choice of a theory T describing a set
ℕ of natural numbers as part of its model M.
This infinite time between theories, will develop as an endless
hierarchy of infinities.
Achilles runs after a turtle; whenever he crosses the distance to
it, the turtle takes a new length ahead.
Let H be the set of formulas of T with one free
variable intended to range over this ℕ, and shorter than (for example)
1000 occurrences of symbols (taken from the finite list of symbols
of T, logical symbols and variables).
Consider the formula of T' with one free variable n
ranging over ℕ, expressed as
∀F∈H, F(n) ⇒
This formula cannot be false on more than one number per formula in
H, which are only finitely many (an explicit bound of their number
can be found). Thus it must be true on some numbers.
If it was equivalent to some formula B∈H, we would get
∀n∈ℕ, B(n) ⇔ (∀F∈H,
F(n) ⇒ (∃k<n, F(k)))
⇒ (∃k<n, B(k))
contradicting the existence of a smallest n on which B is true.
The number 1000 was picked in case translating this formula into
T was complicated, ending up in a big formula B, but still in
H. If it was so complicated that 1000 symbols didn't suffice, we could
try this reasoning starting from a higher number. Since the existence of an
equivalent formula in H would anyway lead to a contradiction,
no number we might pick can ever suffice to find one. This shows the
impossibility to translate such formulas of T' into equivalent formulas of
T, by any method much more efficient than the kind of mere
enumeration suggested above.
Seen from a height, a vehicle gone on a horizontal road approaches
Particles are sent in accelerators closer and closer to the speed of
Can they reach their ends ?
Each example can be seen in two ways:
- the «closed» view, sees a reachable end;
- the «open» view ignores this end, but only sees the movement
towards it, never reaching it.
And in each example, a physical measure of the «cost» to
approach and eventually reach the targeted end, decides its «true»
interpretation, according to whether this cost would be finite or
infinite, which may differ from the first guess of a naive
Each generic theory is «closed», as it can see its model (the ranges
of its variables) as a whole (that is a set in its set theoretical
formulation): by its use of binders over types (or classes), it
«reaches the end» of its model, and thus sees it as «closed». But
any possible framework for it (one-model theory and/or set theory)
escapes this whole.
But the world of mathematics, free from all physical costs and
where objects only play conventional roles, can accept both
As explained in 1.7, set theory has multiple possible models : from
the study of a given universe of sets, we can switch to that of a
larger one with more sets (that we called meta-sets), and new
functions between the new sets.
As this can be repeated endlessly, we need an «open» theory
integrating each universe described by a theory, as a part (past) of
a later universe, forming an endless sequence of growing realities,
with no view of any definite totality. This role of an open theory
will be played by set theory itself, with the way its expressions only bind
variables on sets (1.8).
FR : Temps en
théorie des modèles